In this paper we consider the optimization problem of an agent who wants to maximize the total expected discounted utility from consumption over an infinite horizon. The agent is under obligation to pay a debt at a fixed rate until he/she declares bankruptcy. At that point, after paying a fixed cost, the agent will be able to keep a certain fraction of the present wealth, and the debt will be forgiven. The selection of the bankruptcy time is taken to be at the discretion of the agent. The novelty of this paper is that at the time of bankruptcy the wealth process has a discontinuity, and that the agent continues to invest and consume after bankruptcy. We show that the solution of a free boundary problem satisfying some additional conditions is the value function of the above optimization problem. Particular examples such as the logarithmic and the power utility functions will be provided, and in these cases explicit forms will be given for the optimal bankruptcy time, investment and consumption processes.
The usual tool for modeling bond ratings migration is a discrete, timehomogeneuous Markov chain. Such model assumes that all bonds are homogeneous with respect to their movement behavior among rating categories and that the movement behavior does not change over time. However, among recognized sources of heterogeneity in ratings migration is age of a bond (time elapsed since issuance). It has been observed that young bonds have a lower propensity to change ratings, and thus to default, than more seasoned bonds.The aim of this paper is to introduce a continuous, time-nonhomogeneuous model for bond ratings migration, which also incorporates a simple form of population heterogeneity. The specific form of heterogeneity postulated by the proposed model appears to be suitable for modeling the effect of age of a bond on its propensity to change ratings. This model, called a mover-stayer model, is an extension of a time-nonhomogeneuous Markov chain. This paper derives the maximum likelihood estimators for the parameters of a continuous time mover-stayer model based on a sample of independent continuously monitored histories of the process, and develops the likelihood ratio test for discriminating between the Markov chain and the mover-stayer model. The methods are illustrated using a sample of rating histories of young corporate issuers. For this sample, the likelihood ratio test rejects a Markov chain in favor of a mover-stayer model. For young bonds with lowest rating the default probabilities predicted by the mover-stayer model are substantially lower than those predicted by the Markov chain.
In a single period framework, we develop a supply portfolio risk assessment tool for raw material procurement in the presence of supply risk (owing to contract breaches), demand risk and the spot price risk. Contract breaches are operational risk events that are classified under the "Clients, Products and Business Practices" category of the Basel II framework. We allow for the negative financial impact of intentional long-term fixed price contract breaches to be mitigated by using the spot market. The manufacturer uses the spot market to procure their need in the presence of a contract breach as well as to handle the shortfall/excess in customer demand. We use the CreditRisk + framework, well known in finance literature, to extend the single supplier model to a portfolio of suppliers. This extension enables us to obtain, in the context of supply risk, the entire loss distribution at the portfolio level. In particular, akin to the value-at-risk statistic in finance, one can easily obtain a simple yet effective quantile measure of supply risk, coined as supply-at-risk, for a portfolio of long-term fixed price supply contracts. 1 Furthermore, 29% of the respondents counted the commodity shortages and price fluctuations as an important supply chain risk.
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